Research efforts have been made in recent years for reducing symbol error rates by using soft-outputs for decoded outputs of inner codes of concatenated codes and also for outputs of iterative decoding operations using an iterative decoding method. Studies of decoding methods for minimizing symbol error rates are being made extensively. For example, Bahl, Cock, Jelinek and Raviv, “Optimal decoding of linear codes for minimizing symbol error rate” IEEE Trans. Inf. Theory, vol. IT-20, pp. 284–287, March 1974, describes BCJR algorithm as a method for minimizing the symbol error rate when decoding a predetermined code such as a convolutional code. The BCJR algorithm is adapted to output the likelihood of each symbol instead of outputting each symbol as a result of a decoding operation. Such an output is referred to as soft-output. Now, the BCJR algorithm will be summarily described below. In the following description, it is assumed that a piece of digital information is turned into a convolutional code by means of encoder 201 of a transmitter (not shown) and the output of the encoder is input to a receiver (not shown) by way of a memoryless communication channel 202 having noise and decoded by decoder 203 of the receiver for observation.
Firstly, M states (transition state) representing the contents of the shift register of the encoder 201 are expressed by m (0, 1, . . . , M−1) and the state at clock time t is expressed by St. Also, if a k-bit information is input in a single time slot the input at clock time t is expressed by it=(it1, it2, . . . , itk) and the input system is expressed by I1T=(i1, i2, . . . , iT). If there is a transition from state m′ to state m, the information bit corresponding to the transition is expressed by i(m′, m)=(i1(m′, m), i2(m′, m), . . . , ik(m′, m)). Additionally, if an n-bit code is output in a single time slot, the output at clock time t is expressed by Xt=(xt1, xt2, . . . , xtn) and the output system is expressed by X1T=(x1, x2, . . . , xT). If there is a transition from state m′ to state m, the code bit corresponding to the transition is expressed by X(m′, m)=(x1(m′, m), x2(m′, m), . . . , xn(m′, m)).
The convolutional coding of the encoder 201 starts at state S0=0 and ends at state ST=0 after outputting X1T. The inter-state transition probability Pt(m|m′) of each state is defined by formula (1) below;Pt(m|m′)=Pr{St=m|St−1=m′}  (1),where Pr{A|B} at the right side represents the conditional probability of occurrence of A under the condition of occurrence of B. This transition probability Pt(m|m′) is equal to the probability Pr{it=i} that it is i at clock time t when a transition from state m′ to state m occurs in response to input i.Pt(m|m′)=Pr{i1=i}  (2)
The memoryless communication channel 202 having noise receives X1T as input and output Y1T. If an n-bit value to be received is output in a single time slot, the output at clock time t is expressed by yt=(yt1, yt2, . . . , ytn) and the expression of Y1T=(y1, y2, . . . , yT) is used. Then, the transition probability of the memoryless communication channel 202 having noise is defined by formula (3) below, using the transition probability Pr{yj|xj} of all the symbols for all t(1≦t≦T).
                              Pr          ⁢                      {                                          Y                1                t                            ❘                              X                1                t                                      }                          =                              ∏                          j              =              1                        t                    ⁢                      Pr            ⁢                          {                                                y                  j                                ❘                                  x                  j                                            }                                                          (        3        )            
Now, λtj is defined by formula (4) below. It will be appreciated that λtj of formula (4) below represents the likelihood of the input information at clock time t when Y1T is received and hence the soft-output to be obtained.
                              λ          ij                =                              Pr            ⁢                          {                                                i                  ij                                =                                  1                  ❘                                      Y                    1                    T                                                              }                                            Pr            ⁢                          {                                                i                  ij                                =                                  0                  ❘                                      Y                    1                    T                                                              }                                                          (        4        )            
With the BCJR algorithm, probabilities α1, β1 and γ1 are defined respectively by formulas (5) through (7) below. Note that Pr{A; B} represents the probability of occurrence of both A and B.αt(m)=Pr{St=m; Y1t}  (5)βt(m)=Pr{Yt+1T|St=m}  (6)γt(m′,m)=Pr{St=m; yt|St−1=m′}  (7)
Now, the probabilities αt, βt and γt will be specifically described below by referring to FIG. 2 of the accompanying drawings that illustrates a trellis diagram, or a state transition diagram that can be used in the encoder 201. In FIG. 2, αt−1 corresponds to the passing probability of each state at clock time t−1 that is computed sequentially on a time series basis from state S0=0 at the coding start time by using the received value and βt corresponds to the passing probability of each state at clock time 1 that is computed sequentially on an inverse time series basis from state ST=0 at the coding end time by using the received value, while γt corresponds to the reception probability of the output of each branch that is transferred from a state to another at clock time t as computed on the basis of the received value at clock time t and the input probability.
Then, the soft-output λtj can be expressed by formula (8) below, using these probabilities αt, βt and γt.
                              λ          tj                =                                            ∑                                                                    m                    ′                                    ,                  m                                ⁢                                                                  ⁢                                                                            i                      j                                        ⁡                                          (                                                                        m                          ′                                                ,                        m                                            )                                                        =                  1                                                      ⁢                                                            α                  t                                ⁡                                  (                                      m                    ′                                    )                                            ⁢                                                γ                  t                                ⁡                                  (                                                            m                      ′                                        ,                    m                                    )                                            ⁢                                                β                  t                                ⁡                                  (                  m                  )                                                                                        ∑                                                                    m                    ′                                    ,                  m                                ⁢                                                                  ⁢                                                                            i                      j                                        ⁡                                          (                                                                        m                          ′                                                ,                        m                                            )                                                        =                  0                                                      ⁢                                                            α                  t                                ⁡                                  (                                      m                    ′                                    )                                            ⁢                                                γ                  t                                ⁡                                  (                                                            m                      ′                                        ,                    m                                    )                                            ⁢                                                β                  t                                ⁡                                  (                  m                  )                                                                                        (        8        )            
Meanwhile, equation (9) below holds true for t=1, 2, . . . , T;
                                                        α              t                        ⁡                          (              m              )                                =                                    ∑                                                m                  ′                                =                0                                            M                -                1                                      ⁢                                                            α                                      t                    -                    1                                                  ⁡                                  (                                      m                    ′                                    )                                            ⁢                                                γ                  t                                ⁡                                  (                                                            m                      ′                                        ,                    m                                    )                                                                    ,                            (        9        )            where, α0(0)=1, α0(m)=0(m≠0)
Similarly, equation (10) below holds true for t=1, 2, . . . , T.
                                                        β              t                        ⁡                          (              m              )                                =                                    ∑                                                m                  ′                                =                0                                            M                -                1                                      ⁢                                                            β                                      t                    +                    1                                                  ⁡                                  (                                      m                    ′                                    )                                            ⁢                                                γ                                      t                    +                    1                                                  ⁡                                  (                                      m                    ,                                          m                      ′                                                        )                                            ⁢                                                          ⁢              where                                      ,                                  ⁢                                            β              T                        ⁡                          (              0              )                                =          1                ,                                            β              T                        ⁡                          (              m              )                                =                      0            ⁢                          (                              m                ≠                0                            )                                                          (        10        )            
Likewise, as for yt, equation (11) below holds true for t=1, 2, . . . , T.
                                          γ            t                    ⁡                      (                                          m                ′                            ,              m                        )                          =                  {                                                                                                                                                                                              P                            t                                                    ⁡                                                      (                                                          m                              ❘                                                              m                                ′                                                                                      )                                                                          ·                        Pr                                            ⁢                                              {                                                                              y                            i                                                    ❘                                                      x                            ⁡                                                          (                                                                                                m                                  ′                                                                ,                                m                                                            )                                                                                                      }                                                              =                                                                                                                    Pr                    ⁢                                                                  {                                                                              i                            t                                                    =                                                      i                            ⁡                                                          (                                                                                                m                                  ′                                                                ,                                m                                                            )                                                                                                      }                                            ·                      Pr                                        ⁢                                          {                                                                        y                          i                                                ❘                                                  x                          ⁡                                                      (                                                                                          m                                ′                                                            ,                              m                                                        )                                                                                              }                                                                                                                                                                                                      ⁢                                                                  :                                            ⁢                                                                                                   *                                                ⁢                        1                                                                                                                                                              0                    ⁢                                           :                                        ⁢                                                                                           *                                            ⁢                      2                                                                                            ⁢                                                  :                                                                               *                                ⁢                1                            ⁢                              :                            ⁢                                                          ⁢              for              ⁢                                                          ⁢              transition              ⁢                                                          ⁢              from              ⁢                                                          ⁢                              m                ′                            ⁢                                                          ⁢              to              ⁢                                                          ⁢              m              ⁢                                                          ⁢              at              ⁢                                                          ⁢              input              ⁢                                                          ⁢                              i                ⁢                                                                  :                                                                                                   *                                        ⁢                    2                                    ⁢                                      :                                    ⁢                                                                          ⁢                  for                  ⁢                                                                          ⁢                  non                  ⁢                                      -                                    ⁢                  transition                  ⁢                                                                          ⁢                  from                  ⁢                                                                          ⁢                                      m                    ′                                    ⁢                                                                          ⁢                  to                  ⁢                                                                          ⁢                  m                  ⁢                                                                          ⁢                  at                  ⁢                                                                          ⁢                  input                  ⁢                                                                          ⁢                  i                                                                                        (        11        )            
Therefore, the decoder 203 determines soft-output λt by following the steps shown in FIG. 3 for soft-output decoding, applying the BCJR algorithm.
Referring to FIG. 3, firstly in Step S201, the decoder 203 computes the probability αt(m) and the probability γt(m′, m), using the above equations (9) and (11) respectively, each time it receives γt.
Then in Step S202, after receiving all the system Y1T, the decoder 203 computes the probability βt(m) for each state m at all clock times t, using the above equation (10).
Subsequently in Step S203, the decoder 203 computes the soft-output λt at each clock time t, substituting the probabilities αt, βt and γt obtained respectively in Steps S201 and S202 for the above equation (8).
Thus, the decoder 203 can decodes the soft-output obtained by applying the BCJR algorithm and following the above processing steps.
However, the above described BCJR algorithm requires computations to be made by directly holding probability values and the computations involve a large number of operations of multiplication, which by turn entails a problem of a large volume of computing operation. Robertson, Villebrun and Hoeher, “A comparison of optimal and sub-optimal MAP decoding algorithms operating in the domain”, IEEE Int. Conf. On Communications, pp. 1009–1013, June 1995, describes a technique for reducing the volume of computing operation by using a Max-Log-MAP algorithm and a Log-MAP algorithm (to be respectively referred to as Max-Log-BCJR algorithm and Log-BCJR algorithm hereinafter).
Firstly, the Max-Log-BCJR algorithm will be described. The Max-Log-BCJR algorithm is designed to express the probabilities αt, βt and γt and the soft-output λt in terms of natural logarithm in order to replace the multiplications for computing the probabilities with additions of natural logarithm in a manner as indicated by formula (12) and the additions for the probabilities are approximated by operations for computing maximum values, using formula (13) below. Note that, max (x, y) in formula (13) is a function selecting the larger value of ether x or y.log(ex·ey)=x+y  (12)log(ex+ey)≅max(x,y)  (13)
Now, for the purpose of simplicity, natural logarithm is denoted by I and the values of αt, βt, γt and λt as expressed in the form of natural logarithm are denoted respectively by Iαt, Iβt, Iγt and Iλt as shown in formula (14) below.
                    {                                                                              I                  ⁢                                                                          ⁢                                                            α                      t                                        ⁡                                          (                      m                      )                                                                      =                                  log                  ⁢                                                                          ⁢                                      (                                                                  α                        t                                            ⁡                                              (                        m                        )                                                              )                                                                                                                                            I                  ⁢                                                                          ⁢                                                            β                      t                                        ⁡                                          (                      m                      )                                                                      =                                  log                  ⁢                                                                          ⁢                                      (                                                                  β                        t                                            ⁡                                              (                        m                        )                                                              )                                                                                                                                            I                  ⁢                                                                          ⁢                                                            γ                      t                                        ⁡                                          (                      m                      )                                                                      =                                  log                  ⁢                                                                          ⁢                                      (                                                                  γ                        t                                            ⁡                                              (                        m                        )                                                              )                                                                                                                                            I                  ⁢                                                                          ⁢                                      λ                    t                                                  =                                  log                  ⁢                                                                          ⁢                                      λ                    t                                                                                                          (        14        )            
With the Max-Log-BCJR algorithm, the log likelihoods Iαt, Iβt, Iγt are approximated by formulas (15) through (17) shown below. Note that maximum value max of the right side of formula (15) in state m′ is determined in state m′ where a transition to state m exists. Similarly, maximum value max of the right side of formula (16) in state m′ is determined in state m′ where a transition from state m exists.
                              I          ⁢                                          ⁢                                    α              t                        ⁡                          (              m              )                                      ≅                              max                          m              ′                                ⁢                      (                                          I                ⁢                                                                  ⁢                                                      α                                          t                      -                      1                                                        ⁡                                      (                                          m                      ′                                        )                                                              +                              I                ⁢                                                                  ⁢                                                      γ                    t                                    ⁡                                      (                                                                  m                        ′                                            ,                      m                                        )                                                                        )                                              (        15        )                                          I          ⁢                                          ⁢                                    β              t                        ⁡                          (              m              )                                      ≅                              max                          m              ′                                ⁢                      (                                          I                ⁢                                                                  ⁢                                                      β                                          t                      +                      1                                                        ⁡                                      (                                          m                      ′                                        )                                                              +                              I                ⁢                                                                  ⁢                                                      γ                                          t                      +                      1                                                        ⁡                                      (                                          m                      ,                                              m                        ′                                                              )                                                                        )                                              (        16        )            Iγt(m′,m)=log(Pr{it=i(m′,m)})+log(Pr{yt|x(m′,m)})  (17)
Additionally, with the Max-Log-BCJR algorithm, the log soft-output Iλt is approximated by formula (18) shown below. Note that maximum value max of the first term of the right side in formula (15) is determined in state m′ where a transition to state m exists when the input is equal to “0”.
                                                                        I                ⁢                                                                  ⁢                                  λ                  ij                                            ≅                            ⁢                                                                    max                                                                                            m                          ′                                                ,                        m                                            ⁢                                                                                          ⁢                                                                                                    i                            j                                                    ⁡                                                      (                                                                                          m                                ′                                                            ,                              m                                                        )                                                                          =                        1                                                                              ⁢                                      (                                                                  I                        ⁢                                                                                                  ⁢                                                                              α                                                          t                              -                              1                                                                                ⁡                                                      (                                                          m                              ′                                                        )                                                                                              +                                              I                        ⁢                                                                                                  ⁢                                                                              γ                            t                                                    ⁡                                                      (                                                                                          m                                ′                                                            ,                              m                                                        )                                                                                              +                                              I                        ⁢                                                                                                  ⁢                                                                              β                            t                                                    ⁡                                                      (                            m                            )                                                                                                                )                                                  -                                                                                                      ⁢                                                max                                                                                    m                        ′                                            ,                      m                                        ⁢                                                                                  ⁢                                                                                            i                          j                                                ⁡                                                  (                                                                                    m                              ′                                                        ,                            m                                                    )                                                                    =                      0                                                                      ⁢                                  (                                                            I                      ⁢                                                                                          ⁢                                                                        α                                                      t                            -                            1                                                                          ⁡                                                  (                                                      m                            ′                                                    )                                                                                      +                                          I                      ⁢                                                                                          ⁢                                                                        γ                          t                                                ⁡                                                  (                                                                                    m                              ′                                                        ,                            m                                                    )                                                                                      +                                          I                      ⁢                                                                                          ⁢                                                                        β                          t                                                ⁡                                                  (                          m                          )                                                                                                      )                                                                                        (        18        )            
When the decoder 203 decodes the soft-output, applying the Max-Log-BCJR algorithm, it determines the soft-output λt on the basis of these relationships, by following the steps shown in FIG. 4.
Referring to FIG. 4, firstly in Step S211, the decoder 203 computes the log likelihood Iαt(m) and the log likelihood Iγt(m′, m), using the above equations (15) and (17) respectively, each time it receives yt.
Then in Step S212, after receiving all the system Y1T, the decoder 203 computes the log likelihood Iβt(m) for each state m at all clock times t, using the above equation (16).
Subsequently in Step S213, the decoder 203 computes the log soft-output Iλt at each clock time t, substituting the log likelihoods Iαt, Iβt and Iγt obtained respectively in Steps S211 and S212 for the above equation (18).
Thus, the decoder 203 can decodes the soft-output obtained by applying the Max-Log-BCJR algorithm and following the above processing steps.
Because the Max-Log-BCJR algorithm does not involve multiplications, it can considerably reduce the volume of operation if compared with the BCJR algorithm.
Now, the Log-BCJR algorithm will be described below. The Log-BCJR algorithm is designed to improve the accuracy of approximation of the Max-Log-BCJR algorithm. More specifically, with the Log-BCJR algorithm, the probability addition of the above formula (13) is modified to produce formula (19) below by adding a correction term to make the sum of the logarithmic addition more accurate. The correction will be referred to as log-sum correction hereinafter.log(ex+ey)=max(x, y)+log(1+e−|x−y|)  (19)
The operation of the left side of the above formula (19) is referred to as log-sum operation and, for the sake of convenience, the operator of the log-sum operation is expressed by “#” as shown in formula (20) below by following the numeration system described in S. S. Pietrobon, “Implementation and performance of a turbo/MAP decoder”, Int. J. Satellite Commun., vol. 16, pp. 23–46, January–February 1998 (although “E” is used to denote the operator in the above identified paper). Additionally, the operator of the cumulative addition of log-sum operations is expressed by “#Σ” as shown in formula (21) below (although “E” is used in the above identified paper).x#y=log(ex+ey)  (20)
                              #          ⁢                                    ∑                              i                =                0                                            M                -                1                                      ⁢                          x              i                                      =                  (                                    (                                                …                  ⁡                                      (                                                                  (                                                                              x                            0                                                    ⁢                          #                          ⁢                                                      x                            1                                                                          )                                            ⁢                      #                      ⁢                                              x                        2                                                              )                                                  ⁢                …                            )                        ⁢            #            ⁢                          x                              M                -                1                                              )                                    (        21        )            
The log likelihoods Iαt and Iβt and the log soft-output Iλt of the Log-BCJR algorithm are expressed respectively by formulas (22) through (24) below using these operators. Since the expression of (17) is also used for the log likelihood Iγt here, it will not be described any further.
                              I          ⁢                                          ⁢                                    α              t                        ⁡                          (              m              )                                      =                  #          ⁢                                    ∑                                                m                  ′                                =                0                                            M                -                1                                      ⁢                          (                                                I                  ⁢                                                                          ⁢                                                            α                                              t                        -                        1                                                              ⁡                                          (                                              m                        ′                                            )                                                                      +                                  I                  ⁢                                                                          ⁢                                                            γ                      t                                        ⁡                                          (                                                                        m                          ′                                                ,                        m                                            )                                                                                  )                                                          (        22        )            
                              I          ⁢                                          ⁢                                    β              t                        ⁡                          (              m              )                                      =                  #          ⁢                                    ∑                                                m                  ′                                =                0                                            M                -                1                                      ⁢                          (                                                I                  ⁢                                                                          ⁢                                                            β                                              t                        +                        1                                                              ⁡                                          (                                              m                        ′                                            )                                                                      +                                  I                  ⁢                                                                          ⁢                                                            γ                                              t                        +                        1                                                              ⁡                                          (                                              m                        ,                                                  m                          ′                                                                    )                                                                                  )                                                          (        23        )                                                                                                      I                  ⁢                                                                          ⁢                                      λ                    tj                                                  =                                ⁢                                                      #                    ⁢                                                                  ∑                                                                                                            m                              ′                                                        ,                            m                                                    ⁢                                                                                                          ⁢                                                                                                                    i                                j                                                            ⁡                                                              (                                                                                                      m                                    ′                                                                    ,                                  m                                                                )                                                                                      =                            1                                                                                              ⁢                                              (                                                                              I                            ⁢                                                                                                                  ⁢                                                                                          α                                                                  t                                  -                                  1                                                                                            ⁡                                                              (                                                                  m                                  ′                                                                )                                                                                                              +                                                      I                            ⁢                                                                                                                  ⁢                                                                                          γ                                t                                                            ⁡                                                              (                                                                                                      m                                    ′                                                                    ,                                  m                                                                )                                                                                                              +                                                      I                            ⁢                                                                                                                  ⁢                                                                                          β                                t                                                            ⁡                                                              (                                m                                )                                                                                                                                    )                                                                              -                                                                                                                        ⁢                                  #                  ⁢                                                            ∑                                                                                                    m                            ′                                                    ,                          m                                                ⁢                                                                                                  ⁢                                                                                                            i                              j                                                        ⁡                                                          (                                                                                                m                                  ′                                                                ,                                m                                                            )                                                                                =                          0                                                                                      ⁢                                          (                                                                        I                          ⁢                                                                                                          ⁢                                                                                    α                                                              t                                -                                1                                                                                      ⁡                                                          (                                                              m                                ′                                                            )                                                                                                      +                                                  I                          ⁢                                                                                                          ⁢                                                                                    γ                              t                                                        ⁡                                                          (                                                                                                m                                  ′                                                                ,                                m                                                            )                                                                                                      +                                                  I                          ⁢                                                                                                          ⁢                                                                                    β                              t                                                        ⁡                                                          (                              m                              )                                                                                                                          )                                                                                                          ⁢                                                      (        24        )            
Note that the cumulative addition of log-sum operations in state m′ of the right side of the above formula (22) is conducted in state m′ where a transition to state m exists, whereas the cumulative addition of log-sum operations in state m′ of the right side of the above formula (23) is conducted in state m′ where a transition from state m exists. Furthermore, the cumulative addition of log-sum operations of the first term of the right side in formula (24) is conducted in state m′ where a transition to state m exists when the input is equal to “1”, and the cumulative addition of log-sum operations of the second term is conducted in state m′ where a transition to state m exists when the input is equal to “0”.
When the decoder 203 decodes the soft-output, applying the Log-BCJR algorithm, it determines the soft-output λt on the basis of these relationships, by following the steps shown in FIG. 4.
Referring to FIG. 4, firstly in Step S211, the decoder 203 computes the log likelihood Iαt(m) and the log likelihood Iγt(m′, m), using the above equations (22) and (17) respectively, each time it receives yt.
Then in Step S212, after receiving all the system Y1T, the decoder 203 computes the log likelihood Iβt(m) for each state m at all clock times t, using the above equation (23).
Subsequently in Step S213, the decoder 203 computes the log soft-output Iλt at each clock time t, substituting the log likelihoods Iαt, Iβt and Iγt obtained respectively in Steps S211 and S212 for the above equation (24).
Thus, the decoder 203 can decodes the soft-output obtained by applying the Log-BCJR algorithm and following the above processing steps. Since the correction term, or the second term, of the right side of the above formula (19) is expressed by means of a linear function of variable |x−y|, it is possible to accurately compute probabilities by storing in advance the values that the term can take in a ROM (not shown).
While the Log-BCJR algorithm may entail a volume of operation that is greater than the Max-Log-BCJR algorithm, it does not involve any multiplications and its output is exactly the logarithmic value of the soft-output of the BCJR algorithm if the quantization error is eliminated.
Meanwhile, the above described log likelihood Iγt comprises a priori probability information shown in the first term and probability information (to be referred to as channel value hereinafter) obtained from the received value yt of the second term of the right side of the above equation (17).
With the Max-Log-BCJR algorithm and the Log-BCJR algorithm, it is also possible to use a log likelihood ratio that is a natural log value of the ratio of two probabilities. Then, the log likelihood Iγt can be expressed by formula (25) below. More specifically, the log likelihood Iγt can be expressed by using a priori probability information that is the cumulative sum of multiplication products of the natural log values of the log likelihood ratios of probability Pr{itj=1} and probability Pr{itj=0} and input ij(m′, m), and a channel value that is the cumulative sum of multiplication products of the natural log values of the log likelihood ratios of probability Pr{ytj|xtj=1} and the probability Pr{ytj|xtj=0} and output xj(m′, m).
                                                                        I                ⁢                                                                  ⁢                                  γ                  t                                            =                            ⁢                                                                    ∑                                          j                      =                      1                                        k                                    ⁢                                                                                                              i                          j                                                ⁡                                                  (                                                                                    m                              ′                                                        ,                            m                                                    )                                                                    ·                      log                                        ⁢                                                                                  ⁢                                                                  Pr                        ⁢                                                  {                                                                                    i                              tj                                                        =                            1                                                    }                                                                                            Pr                        ⁢                                                  {                                                                                    i                              tj                                                        =                            0                                                    }                                                                                                                    +                                                                                                      ⁢                                                ∑                                      j                    =                    1                                    n                                ⁢                                                                                                    x                        j                                            ⁡                                              (                                                                              m                            ′                                                    ,                          m                                                )                                                              ·                    log                                    ⁢                                                                          ⁢                                                            Pr                      ⁢                                              {                                                                                                            y                              tj                                                        ❘                                                          x                              tj                                                                                =                          1                                                }                                                                                    Pr                      ⁢                                              {                                                                                                            y                              tj                                                        ❘                                                          x                              tj                                                                                =                          0                                                }                                                                                                                                                    (        25        )            
If it is assumed that additive white Gaussian noise (AWGN) is added to the output xtj of the encoder 201 by the memoryless communication channel 202, the second term of the above formula (25) can be developed in a manner as shown by formula (26) below. Note that “A” in the formula (26) represents the received value ytj that maximizes the probability Pr{ytj|xtj=1} or, in other words, the average value of the normal distribution that the probability density function using the received value ytj as variable follows and “σ2” represents the variance of the normal distribution. The received value ytj that maximizes the probability Pr{ytj|xtj=0} is expressed by “−A”. In reality, “A” and “−A” are the transmission amplitudes of the outputs xtj=1, 0 from the encoder 201.
                                                                        log                ⁢                                                                  ⁢                                                      Pr                    ⁢                                          {                                                                                                    y                            tj                                                    ❘                                                      x                            tj                                                                          =                        1                                            }                                                                            Pr                    ⁢                                          {                                                                                                    y                            tj                                                    ❘                                                      x                            tj                                                                          =                        0                                            }                                                                                  =                              log                ⁢                                                                  ⁢                                                                            1                                                                        2                          ⁢                                                      πσ                            2                                                                                                                ⁢                                          ⅇ                                              -                                                                                                            (                                                                                                y                                  tj                                                                -                                A                                                            )                                                        2                                                                                2                            ⁢                                                          σ                              2                                                                                                                                                                                                      1                                                                                                    2                            ⁢                                                          πσ                              2                                                                                                      ⁢                                                                                                                                        ⁢                                          ⅇ                                              -                                                                                                            (                                                                                                y                                  tj                                                                +                                A                                                            )                                                        2                                                                                2                            ⁢                                                          σ                              2                                                                                                                                                                                                                                                  =                              log                ⁢                                                                  ⁢                                  ⅇ                                      (                                                                                                                        (                                                                                          y                                tj                                                            +                              A                                                        )                                                    2                                                                          2                          ⁢                                                      σ                            2                                                                                              -                                                                                                    (                                                                                          y                                tj                                                            -                              A                                                        )                                                    2                                                                          2                          ⁢                                                      σ                            2                                                                                                                )                                                                                                                          =                                                                                                                  (                                                                              y                            tj                                                    +                          A                                                )                                            2                                                              2                      ⁢                                              σ                        2                                                                              -                                                                                    (                                                                              y                            tj                                                    -                          A                                                )                                            2                                                              2                      ⁢                                              σ                        2                                                                                            =                                                                            2                      ⁢                      A                                                              σ                      2                                                        ⁢                                      y                    tj                                                                                                          (        26        )            
Thus, the decoder 203 can directly obtain the channel value from the received value yt by knowing the characteristics of the memoryless communication channel 202. More specifically, the Iγ computation circuit for computing the log likelihood Iγt of the decoder 203 can be composed of only an adder as shown in FIG. 6. In other words, the decoder 203 can determine the channel value by multiplying the received value ytj by a predetermined coefficient AMP and then the log likelihood Iγt by adding the channel value and a priori probability information (which is denoted by APP in FIG. 6 and will be referred to as a priori probability information APP hereinafter) by means of the Iγ computation circuit. Differently stated, when the decoder 203 needs to multiply the received value ytj by an appropriate coefficient AMP when it determines the log likelihood Iγt.
However, when a decoder 203 is actually installed as hardware, it is difficult to determine the log likelihood Iγt by the arrangement of FIG. 6 because the quantization range is limited.
More specifically, since the coefficient AMP normally takes a value between 0.5 and 10, the distribution of the channel value obtained by multiplying the received value ytj by coefficient AMP varies remarkable relative to the dynamic range of a priori probability information APP. If, for example, a range between 0.5 and 8 is selected for coefficient APP, the range of distribution of the channel value obtained by multiplying the received value ytj by coefficient AMP can be at most 16 times as much as the dynamic range of a priori probability information APP.
Therefore, the decoder 203 cannot be provided with a dynamic range that is wide enough for satisfactorily expressing information on the received value and hence it is difficult to accurately determine the log likelihood Iγt.
Assume that the encoder 201 is one where a plurality of convolutional encoders are concatenated in parallel or in series with interleavers interposed between them for parallel concatenated convolutional coding (to be referred to as PCCC hereinafter) or serially concatenated convolutional coding (to be referred to as SCCC hereinafter) or one where PCCC or SCCC is combined with multi-valued modulation for turbo trellis coded modulation (to be referred to as TTCM hereinafter) or serial concatinated trellis coded modulation (to be referred to as SCTCM hereinafter), whichever approprate. In this case, the decoder 203 is formed by concatenating a plurality of soft-output decoding circuits designed for maximum a poteriori probability (MAP) decoding with interleavers or deinterleavers interposed between them for the purpose of so-called iterative decoding.
Extrinsic information that is the difference between the above a posteriori probability information that corresponds to the above described soft-output or log soft-output and the a priori probability information (APP) is used as a priori probability information APP at the time of iterative decoding. In other words, a priori probability information APP is a pile of received values ytj. When this is taken into consideration, it will not be desirable for the decoder 203 that there are fluctuations in the distribution of the received value ytj relative to the dynamic range of a priori probability information APP.
In other words, it will be desirable for the decoder 203 to make the resolution of a priori probability information APP or the quantization slot of a priori probability information APP variable as a function of the signal to noise ratio (S/N), while maintaining the ratio of the dynamic range of a priori probability information APP to the signal point of the received value ytj to a constant level.
Such a decoder 203 can be realized substantially by employing a configuration as shown in FIG. 8 for the Iγ computation circuit for computing the log likelihood Iγt. Then, the decoder 203 can determine the log likelihood Iγt by adding the quotient obtained by dividing the a priori probability information APP by said coefficient AMP to the received value ytj. With such an arrangement, the decoder 203 can hold the dynamic range of the received value ytj without relaying on the coefficient AMP.
However, if “0.5” is selected for the coefficient AMP, the range that can be used for expressing a priori probability information APP extends between −2×Max and 2×Max relative to the upper limit (Max) and the lower limit (−Max) of the quantization range of the received value ytj, if “8” is selected for the coefficient AMP, the range to be used for expressing a priori probability information APP extends between (−⅛)×Max and (⅛)×Max. In other words, the range that can be used for expressing a priori probability information APP becomes variable. Additionally, the quantization slot of a priori probability information APP also becomes variable. Thus, if a priori probability information APP becomes very fine and goes beyond the allowable range of quantization, it will be no longer possible to express a priori probability information APP.
In this way, when the resolution of a priori probability information APP is modified, there may arise occasions where a priori probability information APP can no longer be expressed to make it impossible to accurately determine the log likelihood Iγt, although a dynamic range can be secured for the received value ytj.